nLab
HOMFLY-PT polynomial

The HOMFLY-PT Polynomial

Idea

The HOMFLY-PT polynomial is a knot and link invariant. Confusingly, there are several variants depending on exactly which relationships are used to define it. All are related by simple substitutions.

Definition

To compute the HOMFLY-PT polynomial, one starts from an oriented link diagram? and uses the following rules:

  1. PP is an isotopy invariant (thus, unchanged by Reidemeister moves).

  2. P(unknot)=1P(\text{unknot}) = 1

  3. Let L +L_+, L L_-, and L 0L_0 be links which are the same except for one part where they differ according to the diagrams below. Then, depending on the choice of variables:

    1. lP(L +)+l 1P(L )+mP(L 0)=0l \cdot P(L_+) + l^{-1} \cdot P(L_-) + m \cdot P(L_0) = 0.
    2. aP(L +)a 1P(L )=zP(L 0)a \cdot P(L_+) - a^{-1} \cdot P(L_-) = z \cdot P(L_0). (Sometimes ν\nu is used instead of aa)
    3. α 1P(L +)αP(L )=zP(L 0)\alpha^{-1} \cdot P(L_+) - \alpha \cdot P(L_-) = z \cdot P(L_0).
    4. Using three variables: xP(L +)+yP(L )+zP(L 0)=0x \cdot P(L_+) + y \cdot P(L_-) + z \cdot P(L_0) = 0.
    L + L L 0 \begin{array}{ccc} \begin{svg}<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0l-57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0l-57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 1.7 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0l57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0l57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \begin{svg}<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0l57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0l57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0l-57 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0l-57 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 1.7 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} & \begin{svg}<svg viewBox="-2.5 -2.5 61.90549 61.90549 " width="62pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" height="62pt"><g transform="translate(0 59) scale(1 -1) translate(0 2.5)"><g stroke="#000"><g fill="#000"><g stroke-width=".4pt"><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m0 0c21 21 21 36 .28 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m0 0c21 21 21 36 .28 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 2 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(-.71 .71 -.71 -.71 2 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"><g stroke-width="5pt"><path d="m57 0c-21 21-21 36 0 57" fill="none"/><g stroke-width="1pt"><g stroke="#f00"><path d="m57 0c-21 21-21 36 0 57" fill="none"/></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke="#f00"><g fill="#f00"><g transform="matrix(.71 .71 -.71 .71 55 55)"><g transform="matrix(1 0 0 1 0 0)"><g stroke-dasharray="none" stroke-dashoffset="0pt"><g stroke-linejoin="miter"><path d="m-7.3 4.5l7.8-4.5-7.8-4.5z"/></g></g></g></g></g></g><g stroke-width="2pt"><g stroke="#fff"><g fill="#fff"></g></g></g></g></g></g></g></g></g></g></svg>\end{svg} \\ L_+ & L_- & L_0 \end{array}

From the rules, one can read off the relationships between the different formulations:

  1. y=α=a 1y = \alpha = a^{-1}
  2. x=α 1=ax = - \alpha^{-1} = -a
  3. a=ila = - i l, l=ial = i a
  4. z=imz = i m, m=izm = - i z.

Properties

The HOMFLY polynomial generalises both the Jones polynomial and the Alexander polynomial (equivalently, the Conway polynomial?).

  • To get the Jones polynomial, make one of the following substitutions:

    1. a=q 1a = q^{-1} and z=q 1/2q 1/2z = q^{1/2} - q^{-1/2}
    2. α=q\alpha = q and z=q 1/2q 1/2z = q^{1/2} - q^{-1/2}
    3. l=iq 1l = i q^{-1} and m=i(q 1/2q 1/2)m = i (q^{-1/2} - q^{1/2})
  • To get the Conway polynomial?, make one of the following substitutions:

    1. a=1a = 1
    2. α=1\alpha = 1
    3. l=il = i, m=izm = -i z
  • To get the Alexander polynomial, make one of the following substitutions:

    1. a=1a = 1, z=q 1/2q 1/2z = q^{1/2} - q^{-1/2}
    2. α=1\alpha = 1, z=q 1/2q 1/2z = q^{1/2} - q^{-1/2}
    3. l=il = i, m=i(q 1/2q 1/2)m = i (q^{-1/2} - q^{1/2})

References

See the wikipedia page for the origin of the name.

Some fairly elementary discussion of the HOMFLY polynomial is given introductory texts such as

The original work was published as

  • P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, and A. Ocneanu. (1985). A New Polynomial Invariant of Knots and Links Bulletin of the American Mathematical Society 12 (2): 239–246.

More recent work includes:

  • A.Mironov, A.Morozov, An.Morozov, Character expansion for HOMFLY polynomials. I. Integrability and difference equations, arxiv/1112.5754

category: knot theory

Revised on December 1, 2012 17:12:38 by Tim Porter (95.147.237.68)